Let us set some global options for all code chunks in this
document.
# Set seed for reproducibility
set.seed(1982)
# Set global options for all code chunks
knitr::opts_chunk$set(
# Disable messages printed by R code chunks
message = FALSE,
# Disable warnings printed by R code chunks
warning = FALSE,
# Show R code within code chunks in output
echo = TRUE,
# Include both R code and its results in output
include = TRUE,
# Evaluate R code chunks
eval = TRUE,
# Enable caching of R code chunks for faster rendering
cache = FALSE,
# Align figures in the center of the output
fig.align = "center",
# Enable retina display for high-resolution figures
retina = 2,
# Show errors in the output instead of stopping rendering
error = TRUE,
# Do not collapse code and output into a single block
collapse = FALSE
)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)
library(plotly)
We want to solve the fractional diffusion equation \[\begin{equation}
\label{eq:maineq}
\partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text {
on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma,
\end{equation}\] where \(u\)
satisfies the Kirchhoff vertex conditions \[\begin{equation}
\label{eq:Kcond}
\left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V:
\sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\}
\end{equation}\]
Let us start by building a graph.
graph <- metric_graph$new(perform_merges = TRUE,
tolerance = list(edge_edge = 1e-3,
vertex_vertex = 1e-3,
edge_vertex = 1e-3))
graph$plot()

graph$build_mesh(h = 0.05)
Let \(\alpha\in(0,2]\) and \(U_h^\tau\) denote the sequence of
approximations of the solution to the weak form of problem \(\eqref{eq:maineq}\) at each time step on a
mesh indexed by \(h\). Let \(z=0\) and \(U^0_h
= P_hu_0\). For \(k=0,\dots,
N-1\), \(U_h^{k+1}\in V_h\)
solves the following scheme \[\begin{align}
\label{system:fully_discrete_scheme}
\langle\delta U_h^{k+1},\phi\rangle +
\mathfrak{a}(U_h^{k+1},\phi) = \langle f^{k+1},\phi\rangle
,\quad\forall\phi\in V_h,
\end{align}\] where \(f^{k+1} =
\displaystyle\dfrac{1}{\tau}\int_{t_k}^{t^{k+1}}f(t)dt\). The
solution can be represented as \[\begin{align*}
U_h^k(s) = \sum_{j=1}^{N_h}u_j^k\psi_j(s)
\end{align*}\] Replacing this into \(\eqref{system:fully_discrete_scheme}\)
yields the following linear system \[\begin{align*}
\sum_{j=1}^{N_h}u_j^{k+1}[(\psi_j,\psi_i)_{L_2(\Gamma)}+
\tau\mathfrak{a}(\psi_j,\psi_i)] =
\sum_{j=1}^{N_h}u_j^{k}(\psi_j,\psi_i)_{L_2(\Gamma)}+\tau(
f^{k+1},\psi_i)_{L_2(\Gamma)}
\end{align*}\] for \(i = 1,\dots,
N_h\). In matrix notation, \[\begin{align}
\label{diff_eq_discrete}
(C+\tau A)U^{k+1} = CU^k+\tau F^{k+1},
\end{align}\] where \(C\) has
entries \(C_{ij} =
(\psi_j,\psi_i)_{L_2(\Gamma)}\), \(A\) has entries \(A_{ij} = \mathfrak{a}(\psi_j,\psi_i)\),
\(U^k\) has entries \(u_j^k\), and \(F^k\) has entries \(( f^{k},\psi_i)_{L_2(\Gamma)}\). By
multiplying both sides by \(A^{-1}\)
and considering its operator-based rational approximation \(P_\ell^{-1}P_r\), we arrive at \((P_rC+\tau P_\ell)U^{k+1} = P_r(CU^k+\tau
F^{k+1})\).
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
# Initial condition
U_0 <- 10*exp(-((x-4)^2 + (y-4)^2))
# Define the time step
time_step <- 0.1
# Define the right-hand side function
fun <- function(t) {return(sin(t)*((x-4)^2 - (y-4)^2))}
# Define the time discretization
time_seq <- seq(0,pi, by = time_step)
# Compute the right-hand side function at each time step
fun_mat <- do.call(cbind, lapply(time_seq, fun))
# Define the parameters
kappa <- 1
L <- kappa^2*C + G
alpha <- 0.8
beta <- alpha/2
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
funF <- C %*% fun_mat
# Precompute the LHS matrix
LHS <- Pr %*% C + time_step * Pl
# Initialize U matrix to store solution at each time step
U_mat <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_mat[, 1] <- U_0
# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
RHS <- Pr %*% (C %*% U_mat[, k] + time_step * funF[, k + 1])
U_mat[, k + 1] <- as.matrix(solve(LHS, RHS))
}
# Plot the initial condition
p_ini <- graph$plot_function(X = U_0,
vertex_size = 1,
type = "plotly",
edge_color = "black",
edge_width = 3,
line_color = "blue",
line_width = 3)
p_ini
# Plot the movie of f
p_f <- graph$plot_movie(fun_mat)
p_f$x$layout$scene$xaxis$range <- range(x)
p_f$x$layout$scene$yaxis$range <- range(y)
p_f$x$layout$scene$zaxis$range <- range(fun_mat)
p_f
# Plot the movie of the solution
p_sol <- graph$plot_movie(U_mat)
p_sol$x$layout$scene$xaxis$range <- range(x)
p_sol$x$layout$scene$yaxis$range <- range(y)
p_sol$x$layout$scene$zaxis$range <- range(U_mat)
p_sol
References
cite_packages(output = "paragraph", out.dir = ".")
We used R version 4.4.3 (R Core Team
2025) and the following R packages: htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.4.16 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and
Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck
et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018;
Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.12.0.9012 (Yuan et al. 2017; Bachl et al. 2019), knitr v.
1.48 (Xie 2014, 2015, 2024), MetricGraph
v. 1.4.1.9000 (Bolin, Simas, and Wallin 2023a,
2023b, 2024, 2025; Bolin et al. 2024), plotly v. 4.10.4 (Sievert 2020), rmarkdown v. 2.28 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and
Riederer 2020; Allaire et al. 2024), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin,
Simas, and Xiong 2024), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).
Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier
Luraschi, Kevin Ushey, Aron Atkins, et al. 2024.
rmarkdown: Dynamic Documents for r.
https://github.com/rstudio/rmarkdown.
Bachl, Fabian E., Finn Lindgren, David L. Borchers, and Janine B.
Illian. 2019.
“inlabru: An
R Package for Bayesian Spatial Modelling from
Ecological Survey Data.” Methods in Ecology and
Evolution 10: 760–66.
https://doi.org/10.1111/2041-210X.13168.
Bakka, Haakon, Håvard Rue, Geir-Arne Fuglstad, Andrea I. Riebler, David
Bolin, Janine Illian, Elias Krainski, Daniel P. Simpson, and Finn K.
Lindgren. 2018.
“Spatial Modelling with INLA:
A Review.” WIRES (Invited Extended Review)
xx (Feb): xx–.
http://arxiv.org/abs/1802.06350.
Bolin, David, and Kristin Kirchner. 2020.
“The Rational
SPDE Approach for Gaussian Random Fields with
General Smoothness.” Journal of Computational and Graphical
Statistics 29 (2): 274–85.
https://doi.org/10.1080/10618600.2019.1665537.
Bolin, David, Mihály Kovács, Vivek Kumar, and Alexandre B. Simas. 2024.
“Regularity and Numerical Approximation of Fractional Elliptic
Differential Equations on Compact Metric Graphs.” Mathematics
of Computation 93 (349): 2439–72.
https://doi.org/10.1090/mcom/3929.
Bolin, David, and Alexandre B. Simas. 2023.
rSPDE: Rational Approximations of Fractional
Stochastic Partial Differential Equations.
https://CRAN.R-project.org/package=rSPDE.
Bolin, David, Alexandre B. Simas, and Jonas Wallin. 2023a.
MetricGraph: Random Fields on Metric Graphs.
https://CRAN.R-project.org/package=MetricGraph.
———. 2023b.
“Statistical Inference for Gaussian Whittle-Matérn
Fields on Metric Graphs.” arXiv Preprint
arXiv:2304.10372.
https://doi.org/10.48550/arXiv.2304.10372.
———. 2024.
“Gaussian Whittle-Matérn Fields on Metric
Graphs.” Bernoulli 30 (2): 1611–39.
https://doi.org/10.3150/23-BEJ1647.
———. 2025.
“Markov Properties of Gaussian Random Fields on Compact
Metric Graphs.” Bernoulli.
https://doi.org/10.48550/arXiv.2304.03190.
Bolin, David, Alexandre B. Simas, and Zhen Xiong. 2024.
“Covariance-Based Rational Approximations of Fractional SPDEs for
Computationally Efficient Bayesian Inference.” Journal of
Computational and Graphical Statistics 33 (1): 64–74.
https://doi.org/10.1080/10618600.2023.2231051.
De Coninck, Arne, Bernard De Baets, Drosos Kourounis, Fabio Verbosio,
Olaf Schenk, Steven Maenhout, and Jan Fostier. 2016.
“Needles: Toward Large-Scale Genomic Prediction with
Marker-by-Environment Interaction.” Genetics 203 (1):
543–55.
https://doi.org/10.1534/genetics.115.179887.
Kourounis, D., A. Fuchs, and O. Schenk. 2018.
“Towards the Next
Generation of Multiperiod Optimal Power Flow Solvers.” IEEE
Transactions on Power Systems PP (99): 1–10.
https://doi.org/10.1109/TPWRS.2017.2789187.
Lindgren, Finn, and Håvard Rue. 2015.
“Bayesian Spatial Modelling
with R-INLA.” Journal of
Statistical Software 63 (19): 1–25.
http://www.jstatsoft.org/v63/i19/.
Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An
Explicit Link Between Gaussian Fields and
Gaussian Markov Random Fields: The Stochastic
Partial Differential Equation Approach (with Discussion).”
Journal of the Royal Statistical Society B 73 (4): 423–98.
Martins, Thiago G., Daniel Simpson, Finn Lindgren, and Håvard Rue. 2013.
“Bayesian Computing with INLA: New
Features.” Computational Statistics and Data Analysis
67: 68–83.
R Core Team. 2025.
R: A Language and Environment for
Statistical Computing. Vienna, Austria: R Foundation for
Statistical Computing.
https://www.R-project.org/.
Rue, Håvard, Sara Martino, and Nicholas Chopin. 2009. “Approximate
Bayesian Inference for Latent Gaussian Models
Using Integrated Nested Laplace Approximations (with
Discussion).” Journal of the Royal Statistical Society B
71: 319–92.
Rue, Håvard, Andrea I. Riebler, Sigrunn H. Sørbye, Janine B. Illian,
Daniel P. Simpson, and Finn K. Lindgren. 2017.
“Bayesian Computing
with INLA: A Review.” Annual
Reviews of Statistics and Its Applications 4 (March): 395–421.
http://arxiv.org/abs/1604.00860.
Sievert, Carson. 2020.
Interactive Web-Based Data Visualization with
r, Plotly, and Shiny. Chapman; Hall/CRC.
https://plotly-r.com.
Verbosio, Fabio, Arne De Coninck, Drosos Kourounis, and Olaf Schenk.
2017.
“Enhancing the Scalability of Selected Inversion
Factorization Algorithms in Genomic Prediction.” Journal of
Computational Science 22 (Supplement C): 99–108.
https://doi.org/10.1016/j.jocs.2017.08.013.
Xie, Yihui. 2014. “knitr: A
Comprehensive Tool for Reproducible Research in R.”
In Implementing Reproducible Computational Research, edited by
Victoria Stodden, Friedrich Leisch, and Roger D. Peng. Chapman;
Hall/CRC.
———. 2015.
Dynamic Documents with R and Knitr. 2nd
ed. Boca Raton, Florida: Chapman; Hall/CRC.
https://yihui.org/knitr/.
———. 2024.
knitr: A General-Purpose
Package for Dynamic Report Generation in r.
https://yihui.org/knitr/.
Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018.
R Markdown:
The Definitive Guide. Boca Raton, Florida: Chapman; Hall/CRC.
https://bookdown.org/yihui/rmarkdown.
Xie, Yihui, Christophe Dervieux, and Emily Riederer. 2020.
R
Markdown Cookbook. Boca Raton, Florida: Chapman; Hall/CRC.
https://bookdown.org/yihui/rmarkdown-cookbook.
Yuan, Yuan, Bachl, Fabian E., Lindgren, Finn, Borchers, et al. 2017.
“Point Process Models for Spatio-Temporal Distance Sampling Data
from a Large-Scale Survey of Blue Whales.” Ann. Appl.
Stat. 11 (4): 2270–97.
https://doi.org/10.1214/17-AOAS1078.
---
title: "Solving a parabolic equation"
date: "Created: 20-04-2025. Last modified: `r format(Sys.time(), '%d-%m-%Y.')`"
output:
  html_document:
    mathjax: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    highlight: pygments
    theme: flatly
    code_folding: show # class.source = "fold-hide" to hide code and add a button to show it
    df_print: paged
    toc: true
    toc_float:
      collapsed: true
      smooth_scroll: true
    number_sections: false
    fig_caption: true
    code_download: true
always_allow_html: true
bibliography: 
  - references.bib
  - grateful-refs.bib
header-includes:
  - \newcommand{\ar}{\mathbb{R}}
  - \newcommand{\llav}[1]{\left\{#1\right\}}
  - \newcommand{\pare}[1]{\left(#1\right)}
  - \newcommand{\Ncal}{\mathcal{N}}
  - \newcommand{\Vcal}{\mathcal{V}}
  - \newcommand{\Ecal}{\mathcal{E}}
  - \newcommand{\Wcal}{\mathcal{W}}
---

```{r xaringanExtra-clipboard, echo = FALSE}
htmltools::tagList(
  xaringanExtra::use_clipboard(
    button_text = "<i class=\"fa-solid fa-clipboard\" style=\"color: #00008B\"></i>",
    success_text = "<i class=\"fa fa-check\" style=\"color: #90BE6D\"></i>",
    error_text = "<i class=\"fa fa-times-circle\" style=\"color: #F94144\"></i>"
  ),
  rmarkdown::html_dependency_font_awesome()
)
```


```{css, echo = FALSE}
body .main-container {
  max-width: 100% !important;
  width: 100% !important;
}
body {
  max-width: 100% !important;
}

body, td {
   font-size: 16px;
}
code.r{
  font-size: 14px;
}
pre {
  font-size: 14px
}
.custom-box {
  background-color: #f5f7fa; /* Light grey-blue background */
  border-color: #e1e8ed; /* Light border color */
  color: #2c3e50; /* Dark text color */
  padding: 15px; /* Padding inside the box */
  border-radius: 5px; /* Rounded corners */
  margin-bottom: 20px; /* Spacing below the box */
}
.caption {
  margin: auto;
  text-align: center;
  margin-bottom: 20px; /* Spacing below the box */
}
```


Let us set some global options for all code chunks in this document.


```{r}
# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
```




```{r}
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)

library(plotly)
```


We want to solve the fractional diffusion equation
\begin{equation}
\label{eq:maineq}
    \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma,
\end{equation}
where $u$ satisfies the Kirchhoff vertex conditions
\begin{equation}
\label{eq:Kcond}
    \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\}
\end{equation}

Let us start by building a graph.

```{r}
graph <- metric_graph$new(perform_merges = TRUE, 
                          tolerance = list(edge_edge = 1e-3, 
                                           vertex_vertex = 1e-3, 
                                           edge_vertex = 1e-3))
graph$plot()
graph$build_mesh(h = 0.05)
```

Let $\alpha\in(0,2]$ and $U_h^\tau$ denote the sequence of approximations of the solution to the weak form of problem \eqref{eq:maineq} at each time step on a mesh indexed by $h$. Let $z=0$ and $U^0_h = P_hu_0$. For $k=0,\dots, N-1$, $U_h^{k+1}\in V_h$ solves the following scheme
\begin{align}
\label{system:fully_discrete_scheme}
        \langle\delta U_h^{k+1},\phi\rangle + \mathfrak{a}(U_h^{k+1},\phi) = \langle f^{k+1},\phi\rangle ,\quad\forall\phi\in V_h,
\end{align}
where $f^{k+1} = \displaystyle\dfrac{1}{\tau}\int_{t_k}^{t^{k+1}}f(t)dt$.
The solution can be represented as 
\begin{align*}
    U_h^k(s) =  \sum_{j=1}^{N_h}u_j^k\psi_j(s)
\end{align*}
Replacing this into \eqref{system:fully_discrete_scheme} yields the following linear system
\begin{align*}
    \sum_{j=1}^{N_h}u_j^{k+1}[(\psi_j,\psi_i)_{L_2(\Gamma)}+ \tau\mathfrak{a}(\psi_j,\psi_i)] = \sum_{j=1}^{N_h}u_j^{k}(\psi_j,\psi_i)_{L_2(\Gamma)}+\tau( f^{k+1},\psi_i)_{L_2(\Gamma)}
\end{align*}
for $i = 1,\dots, N_h$. In matrix notation,
\begin{align}
\label{diff_eq_discrete}
    (C+\tau A)U^{k+1} = CU^k+\tau F^{k+1},
\end{align}
where $C$ has entries $C_{ij} = (\psi_j,\psi_i)_{L_2(\Gamma)}$, $A$ has entries $A_{ij} = \mathfrak{a}(\psi_j,\psi_i)$, $U^k$ has entries $u_j^k$, and $F^k$ has entries $( f^{k},\psi_i)_{L_2(\Gamma)}$. By multiplying both sides by $A^{-1}$ and considering its operator-based rational approximation $P_\ell^{-1}P_r$, we arrive at $(P_rC+\tau P_\ell)U^{k+1} = P_r(CU^k+\tau F^{k+1})$.

```{r}
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
# Initial condition
U_0 <- 10*exp(-((x-4)^2 + (y-4)^2))
# Define the time step
time_step <- 0.1
# Define the right-hand side function
fun <- function(t) {return(sin(t)*((x-4)^2 - (y-4)^2))}
# Define the time discretization
time_seq <- seq(0,pi, by = time_step)
# Compute the right-hand side function at each time step
fun_mat <- do.call(cbind, lapply(time_seq, fun))
# Define the parameters
kappa <- 1
L <- kappa^2*C + G
alpha <- 0.8
beta <- alpha/2
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
funF <- C %*% fun_mat 
```


```{r}
# Precompute the LHS matrix
LHS <- Pr %*% C + time_step * Pl
# Initialize U matrix to store solution at each time step
U_mat <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_mat[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  RHS <- Pr %*% (C %*% U_mat[, k] + time_step * funF[, k + 1])
  U_mat[, k + 1] <- as.matrix(solve(LHS, RHS))
}
```


```{r}
# Plot the initial condition
p_ini <- graph$plot_function(X = U_0, 
                             vertex_size = 1, 
                             type = "plotly", 
                             edge_color = "black", 
                             edge_width = 3, 
                             line_color = "blue", 
                             line_width = 3)
p_ini
# Plot the movie of f
p_f <- graph$plot_movie(fun_mat)
p_f$x$layout$scene$xaxis$range <- range(x)
p_f$x$layout$scene$yaxis$range <- range(y)
p_f$x$layout$scene$zaxis$range <- range(fun_mat)
p_f
# Plot the movie of the solution
p_sol <- graph$plot_movie(U_mat)
p_sol$x$layout$scene$xaxis$range <- range(x)
p_sol$x$layout$scene$yaxis$range <- range(y)
p_sol$x$layout$scene$zaxis$range <- range(U_mat)
p_sol
```



# References

```{r}
cite_packages(output = "paragraph", out.dir = ".")
```
